# EM3.op: Equation of mixed model for multi-kernel using other packages... In RAINBOWR: Genome-Wide Association Study with SNP-Set Methods

## Description

This function solves the following multi-kernel linear mixed effects model using MMEst function in 'MM4LMM' package, lmm.aireml or lmm.diago functions in 'gaston' package, or EM3.cpp function in 'RAINBOWR' package.

y = X β + ∑ _{l=1} ^ {L} Z _ {l} u _ {l} + ε

where Var[y] = ∑ _{l=1} ^ {L} Z _ {l} K _ {l} Z _ {l}' σ _ {l} ^ 2 + I σ _ {e} ^ {2}.

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 EM3.op( y, X0 = NULL, ZETA, eigen.G = NULL, package = "gaston", tol = NULL, n.core = 1, REML = TRUE, pred = TRUE, return.u.always = TRUE, return.u.each = TRUE, return.Hinv = TRUE ) 

## Arguments

 y A n \times 1 vector. A vector of phenotypic values should be used. NA is allowed. X0 A n \times p matrix. You should assign mean vector (rep(1, n)) and covariates. NA is not allowed. ZETA A list of variance matrices and its design matrices of random effects. You can use more than one kernel matrix. For example, ZETA = list(A = list(Z = Z.A, K = K.A), D = list(Z = Z.D, K = K.D)) (A for additive, D for dominance) Please set names of lists "Z" and "K"! eigen.G A list with $valuesEigen values$vectorsEigen vectors The result of the eigen decompsition of G = ZKZ'. You can use "spectralG.cpp" function in RAINBOWR. If this argument is NULL, the eigen decomposition will be performed in this function. We recommend you assign the result of the eigen decomposition beforehand for time saving. package Package name to be used in this function. We only offer the following three packages: "RAINBOWR", "MM4LMM" and "gaston". Default package is 'gaston'. tol The tolerance for detecting linear dependencies in the columns of G = ZKZ'. Eigen vectors whose eigen values are less than "tol" argument will be omitted from results. If tol is NULL, top 'n' eigen values will be effective. n.core Setting n.core > 1 will enable parallel execution on a machine with multiple cores (only for 'MM4LMM'). REML You can choose which method you will use, "REML" or "ML". If REML = TRUE, you will perform "REML", and if REML = FALSE, you will perform "ML". pred If TRUE, the fitting values of y is returned. return.u.always When using the "gaston" package with missing values or using the "MM4LMM" package (with/without missings), computing BLUP will take some time in addition to solving the mixed-effects model. You can choose whether BLUP ('u'; u) will be returned or not. return.u.each If TRUE, the function also computes each BLUP corresponding to different kernels (when solving multi-kernel mixed-effects model). It takes additional time compared to the one with 'return.u.each = FALSE'. return.Hinv If TRUE, H ^ {-1} = (Var[y] / ∑ _{l=1} ^ {L} σ _ {l} ^ 2) ^ {-1} will be computed. It also returns V ^ {-1} = (Var[y]) ^ {-1}. It will take some time in addition to solving the mixed-effects model.

## Value

$y.pred The fitting values of y y = Xβ + Zu$Vu

Estimator for σ^2_u, all of the genetic variance

$Ve Estimator for σ^2_e$beta

BLUE(β)

$u BLUP(Sum of Zu)$u.each

BLUP(Each u)

$weights The proportion of each genetic variance (corresponding to each kernel of ZETA) to Vu$LL

Maximized log-likelihood (full or restricted, depending on method)

$Vinv The inverse of V = Vu \times ZKZ' + Ve \times I$Hinv

The inverse of H = ZKZ' + λ I

## References

Kang, H.M. et al. (2008) Efficient Control of Population Structure in Model Organism Association Mapping. Genetics. 178(3): 1709-1723.

Zhou, X. and Stephens, M. (2012) Genome-wide efficient mixed-model analysis for association studies. Nat Genet. 44(7): 821-824.

Johnson, D. L., & Thompson, R. (1995). Restricted maximum likelihood estimation of variance components for univariate animal models using sparse matrix techniques and average information. Journal of dairy science, 78(2), 449-456.

Hunter, D. R., & Lange, K. (2004). A tutorial on MM algorithms. The American Statistician, 58(1), 30-37.

Zhou, H., Hu, L., Zhou, J., & Lange, K. (2015). MM algorithms for variance components models. arXiv preprint arXiv:1509.07426.

Gilmour, A. R., Thompson, R., & Cullis, B. R. (1995), Average information REML: an efficient algorithm for variance parameter estimation in linear mixed models, Biometrics, 1440-1450.

MMEst, lmm.aireml, lmm.diago